Statistical rethinking homework solutions

INLA book

library(tidyverse)
library(rethinking)
library(dagitty)
library(INLA)
library(knitr)
library(stringr)

Intro to linear prediction from Statistical Rethinking 2nd edition Chapter 4.

Finding the posterior distribution

Bayesian updating will allow us to consider every possible combination of values for μ and σ and to score each combination by its relative plausibility, in light of the data. These relative plausibilities are the posterior probabilities of each combination of values μ, σ. Posterior plausibility provides a measure of the logical compatibility of each possible distribution with the data and model.

The thing to worry about is keeping in mind that the “estimate” here will be the entire posterior distribution, not any point within it. And as a result, the posterior distribution will be a distribution of Gaussian distributions. Yes, a distribution of distributions.

The prior for μ is a broad Gaussian prior, centered on 178cm, with 95% of probability between 178 ± 40.

The weights that interest us are all adult weights, so we can analyze only the adults and make an okay linear approximation.

data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ] 

xbar <- mean(d2$weight) 

The golem is assuming that the average height (not each individual height) is almost certainly between 140 cm and 220 cm

#plot mean prior
 curve( dnorm( x , 178 , 20 ) , from=100 , to=250 )

The σ prior is a truly flat prior, a uniform one, that functions just to constrain σ to have positive probability between zero and 50cm. A standard deviation of 50cm would imply that 95% of individual heights lie within 100cm of the average height. That’s a very large range.

#plot sd prior
  curve( dunif( x , 0 , 50 ) , from=-10 , to=60 )

The prior predictive simulation is an essential part of your modeling. Once you’ve chosen priors for h, μ, and σ, these imply a joint prior distribution of individual heights. By simulating from this distribution, you can see what your choices imply about observable height. This helps you diagnose bad choices. Lots of conventional choices are indeed bad ones, and we’ll be able to see this through prior predictive simulations.

Okay, so how to do this? You can quickly simulate heights by sampling from the prior.

sample_mu <- rnorm( 1e4 , 178 , 20 )
sample_sigma <- runif( 1e4 , 0 , 50 )
prior_h <- rnorm( 1e4 , sample_mu , sample_sigma )
#prior_h 
dens( prior_h )

This is the expected distribution of heights, averaged over the prior. Notice that the prior probability distribution of height is not itself Gaussian. This is okay. The distribution you see is not an empirical expectation, but rather the distribution of relative plausibilities of different heights, before seeing the data.

μ i = α + β ( x i − ̄x )

What this tells the regression golem is that you are asking two questions about the mean of the outcome.

  1. What is the expected height when xi = ̄x? The parameter α answers this question, because when xi = ̄x, μi = α. For this reason, α is often called the intercept. But we should think not in terms of some abstract line, but rather in terms of the meaning with respect to the observable variables.

  2. What is the change in expected height, when xi changes by 1 unit? The parameter β answers this question. It is often called a “slope,” again because of the abstract line. Better to think of it as a rate of change in expectation. Jointly these two parameters ask the golem to find a line that relates x to h, a line that passes through α when xi = ̄x and has slope β. That is a task that golems are very good at. It’s up to you, though, to be sure it’s a good question.

The goal is to simulate heights from the model, using only the priors. First, let’s consider a range of weight values to simulate over. The range of observed weights will do fine. Then we need to simulate a bunch of lines, the lines implied by the priors for α and β. Now we have 100 pairs of α and β values. Now to plot the lines:

set.seed(2971)
N <- 100                   # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rnorm( N , 0 , 10 )


plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
    xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
    from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
    col=col.alpha("black",0.2) )

If the logarithm of β is normal, then β itself is strictly positive. The reason is that exp(x) is greater than zero for any real number x. This is the reason that Log-Normal priors are commonplace. They are an easy way to enforce positive relationships

set.seed(2971)
N <- 100                   # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rlnorm( N , 0 , 1 )


plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
    xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
    from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
    col=col.alpha("black",0.2) )

Overthinking: Logs and exps, oh my. My experience is that many natural and social scientists have naturally forgotten whatever they once knew about logarithms. Logarithms appear all the time in applied statistics. You can usefully think of y = log(x) as assigning to y the order of magnitude of x. The function x = exp(y) is the reverse, turning a magnitude into a value. These definitions will make a mathematician shriek. But much of our computational work relies only on these intuitions. These definitions allow the Log-Normal prior for β to be coded another way. Instead of defining a parameter β, we define a parameter that is the logarithm of β and then assign it a normal distribution. Then we can reverse the logarithm inside the linear model. It looks like this:

m4.3b <- quap( alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + exp(log_b)*( weight - xbar ), a ~ dnorm( 178 , 20 ) ,
log_b ~ dnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
) , data=d2 )

Note the exp(log_b) in the definition of mu. This is the same model as m4.3. It will make the same predictions. But instead of β in the posterior distribution, you get log(β). It is easy to translate between the two, because β = exp(log(β)). In code form: b <- exp(log_b).

Interpreting the posterior distribution. One trouble with statistical models is that they are hard to understand. Once you’ve fit the model, it can only report posterior distribution. This is the right answer to the question you asked. But it’s your responsibility to process the answer and make sense of it.

m4.3 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d2 )

precis(m4.3)
##              mean        sd        5.5%       94.5%
## a     154.6013713 0.2703075 154.1693677 155.0333748
## b       0.9032763 0.0419236   0.8362742   0.9702783
## sigma   5.0718775 0.1911545   4.7663757   5.3773792

The first row gives the quadratic approximation for α, the second the approximation for β, and the third approximation for σ. Let’s try to make some sense of them.

Let’s focus on b (β), because it’s the new parameter. Since β is a slope, the value 0.90 can be read as a person 1 kg heavier is expected to be 0.90 cm taller. 89% of the posterior probability lies between 0.84 and 0.97. That suggests that β values close to zero or greatly above one are highly incompatible with these data and this model. It is most certainly not evidence that the relationship between weight and height is linear, because the model only considered lines. It just says that, if you are committed to a line, then lines with a slope around 0.9 are plausible ones.

You can see the covariances among the parameters with vcov:

round( vcov( m4.3 ) , 3 )
##           a     b sigma
## a     0.073 0.000 0.000
## b     0.000 0.002 0.000
## sigma 0.000 0.000 0.037
# shows both the marginal posteriors and the covariance.
pairs(m4.3)

Very little covariation among the parameters in this case. The lack of covariance among the parameters results from centering.

Plotting posterior inference against the data. It’s almost always much more useful to plot the posterior inference against the data. Not only does plotting help in interpreting the posterior, but it also provides an informal check on model assumptions. When the model’s predictions don’t come close to key observations or patterns in the plotted data, then you might suspect the model either did not fit correctly or is rather badly specified. But even if you only treat plots as a way to help in interpreting the posterior, they are invaluable.

Each point in this plot is a single individual. The black line is defined by the mean slope β and mean intercept α = the posterior mean line. It looks highly plausible. But there an infinite number of other highly plausible lines near it. Let’s draw those too.

plot( height ~ weight , data=d2 , col=rangi2 )
post <- extract.samples( m4.3 )
a_map <- mean(post$a)
b_map <- mean(post$b)
curve( a_map + b_map*(x - xbar) , add=TRUE )

post <- extract.samples( m4.3 ) = Each row is a correlated random sample from the joint posterior of all three parameters, using the covariances provided by vcov(m4.3). The paired values of a and b on each row define a line. The average of very many of these lines is the posterior mean line. But the scatter around that average is meaningful, because it alters our confidence in the relationship between the predictor and the outcome.

Let’s display a bunch of these lines, so you can see the scatter. This lesson will be easier to appreciate, if we use only some of the data to begin. Then you can see how adding in more data changes the scatter of the lines. So we’ll begin with just the first 10 cases in d2. The following code extracts the first 10 cases and re-estimates the model:

N <- 10
dN <- d2[ 1:N , ]
mN <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b*( weight - mean(weight) ) ,
        a ~ dnorm( 178 , 20 ) ,
        b ~ dlnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
) , data=dN )

Now let’s plot 20 of these lines, to see what the uncertainty looks like.

# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )

# display raw data and sample size
plot( dN$weight , dN$height ,
    xlim=range(d2$weight) , ylim=range(d2$height) ,
    col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))

# plot the lines, with transparency
for ( i in 1:20 )
    curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
        col=col.alpha("black",0.3) , add=TRUE )

Increae the amounts of data. Notice that the cloud of regression lines grows more compact as the sample size increases. This is a result of the model growing more confident about the location of the mean.

N <- 352
dN <- d2[ 1:N , ]
mN <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b*( weight - mean(weight) ) ,
        a ~ dnorm( 178 , 20 ) ,
        b ~ dlnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
) , data=dN )

# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )

# display raw data and sample size
plot( dN$weight , dN$height ,
    xlim=range(d2$weight) , ylim=range(d2$height) ,
    col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))

# plot the lines, with transparency
for ( i in 1:20 )
    curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
        col=col.alpha("black",0.3) , add=TRUE )

Focus for the moment on a single weight value, say 50 kilograms. You can quickly make a list of 10,000 values of μ (height)for an individual who weighs 50 kilograms, by using your samples from the posterior.

μ i = α + β ( x i − ̄x ) The value of xi in this case is 50.

mu_at_50 is a vector of predicted means, one for each random sample from the posterior. Since joint a and b went into computing each, the variation across those means incorporates the uncertainty in and correlation between both parameters. It might be helpful at this point to actually plot the density for this vector of means.

Since the components of μ have distributions, so too does μ. And since the distributions of α and β are Gaussian, so to is the distribution of μ (adding Gaussian distributions always produces a Gaussian distribution). Since the posterior for μ is a distribution, you can find intervals for it, just like for any posterior distribution. The central 89% of the ways for the model to produce the data place the average height between about 159 cm and 160 cm (conditional on the model and data), assuming the weight is 50 kg.

post <- extract.samples( m4.3 )
mu_at_50 <- post$a + post$b * ( 50 - xbar )

dens( mu_at_50 , col=rangi2 , lwd=2 , xlab="mu|weight=50" )

PI( mu_at_50 , prob=0.89 )
##       5%      94% 
## 158.5677 159.6695

That’s good so far, but we need to repeat the above calculation for every weight value on the horizontal axis, not just when it is 50 kg. We want to draw 89% intervals around the average slope.

This is made simple by strategic use of the link function, a part of the rethinking package. What link will do is take your quap approximation, sample from the posterior distribution, and then compute μ for each case in the data and sample from the posterior distribution. Here’s what it looks like for the data you used to fit the model:

mu <- link( m4.3 )
str(mu)
##  num [1:1000, 1:352] 157 157 157 157 157 ...

You end up with a big matrix of values of μ. Each row is a sample from the posterior distribu- tion. There are 352 rows in d2, corresponding to 352 individuals. So there are 352 columns in the matrix mu above. link takes 1000 samples of the posterior dist. for every value in the data (of weight in this case).

this is what the rethinking::sim function does

#extract samples automatically extracts 1000 from posterior dist
post <- extract.samples(m4.3)

weight.seq <- 25:70

sim.function <- function(weight)
    rnorm(
        n=nrow(post) ,
        mean=post$a + post$b*( weight - xbar ) ,
        sd=post$sigma )

sim.height <- sapply( weight.seq , sim.function )

##compute the mean of each column (dimension “2”) of the matrix mu.
height.interval <- apply(sim.height, 2, quantile, c( 0.05 , 0.94 ))

#example of compatibility interval for 1st col (first weight)
#quantile(sim.height[,1],c( 0.05 , 0.94 ) )

# plot raw data
plot( height ~ weight , d2 , col=col.alpha(rangi2,0.5) )
# draw MAP line
lines( weight.seq , mu.mean )
# draw HPDI region for line
shade( mu.PI , weight.seq )
# draw PI region for simulated heights
shade( height.interval, weight.seq )

Rethinking: Two kinds of uncertainty. In the procedure above, we encountered both uncertainty in parameter values and uncertainty in a sampling process. These are distinct concepts, even though they are processed much the same way and end up blended together in the posterior predictive simu- lation. The posterior distribution is a ranking of the relative plausibilities of every possible combina- tion of parameter values. The distribution of simulated outcomes, like height, is instead a distribution that includes sampling variation from some process that generates Gaussian random variables. This sampling variation is still a model assumption. It’s no more or less objective than the posterior distribution. Both kinds of uncertainty matter, at least sometimes. But it’s important to keep them straight, because they depend upon different model assumptions.

HOMEWORK 2

1.

The weights listed below were recorded in the !Kung census, but heights were not recorded for these individuals. Provide predicted heights and 89% compatibility intervals for each of these individuals, using model-based predictions.

ind <- 1:5
weight <- c(45, 40, 65, 31, 53)
expected_height <- NA
interval <- NA
kung <- bind_cols(ind, weight, expected_height, interval)
## New names:
## * NA -> ...1
## * NA -> ...2
## * NA -> ...3
## * NA -> ...4
colnames(kung) <- c("individual", "weight", "expected_height", "89%interval")
kable(kung)
individual weight expected_height 89%interval
1 45 NA NA
2 40 NA NA
3 65 NA NA
4 31 NA NA
5 53 NA NA

1. rethinking

m4.3 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d2 )

precis(m4.3)
##              mean         sd        5.5%       94.5%
## a     154.6013672 0.27030763 154.1693634 155.0333710
## b       0.9032809 0.04192363   0.8362788   0.9702829
## sigma   5.0718803 0.19115473   4.7663781   5.3773825

Now we need posterior predictions for each case in the table. Easiest way to do this is to use sim. We need sim, not just link, because we are trying to predict an individual’s height. So the relevant compatibility interval includes the Gaussian variance from sigma. If you provided only the compatibility interval for μ, that’s okay. But be sure you understand the difference.

Solution using rethinking functions

dat <-  data.frame(weight= c(45, 40, 65, 31, 53))
h_sim <- sim( m4.3 , data=dat )
Eh <- apply(h_sim,2,mean)
h_ci <- apply(h_sim,2,PI,prob=0.89)

dat$Eh <- Eh 
dat$L89 <- h_ci[1,] 
dat$U89 <- h_ci[2,] 
round(dat,1)
##   weight    Eh   L89   U89
## 1     45 154.6 146.6 163.0
## 2     40 149.9 141.8 157.9
## 3     65 172.7 164.5 181.1
## 4     31 141.9 133.6 150.2
## 5     53 162.1 154.2 169.8

Solution using base r functions

weight= c(45, 40, 65, 31, 53)

sim.hw2.fun <- function(weight, model) {
  post.hw2 = extract.samples(model)
  rnorm(
    n= nrow(post),
    mean= post$a + post$b*(weight - xbar),
    sd= post$sigma)
}

sim.hw2 <- sapply(weight, sim.hw2.fun , m4.3)

hw2.mean <- apply(sim.hw2,2, mean)
hw2.ci <- apply(sim.hw2,2, quantile, c(0.05, 0.95))

hw2.1 <- data.frame(weight= c(45, 40, 65, 31, 53)) %>% 
  mutate(expected_height= hw2.mean, 
         LCI= hw2.ci[1], 
         UCI = hw2.ci[2])

print(hw2.1)
##   weight expected_height      LCI      UCI
## 1     45        154.5601 146.0763 162.9388
## 2     40        150.1655 146.0763 162.9388
## 3     65        172.7602 146.0763 162.9388
## 4     31        141.9387 146.0763 162.9388
## 5     53        161.8957 146.0763 162.9388

1. inla

https://www.flutterbys.com.au/stats/tut/tut12.10.html

https://people.bath.ac.uk/jjf23/brinla/hubble.html#inla-weakly-or-non-informative

https://people.bath.ac.uk/jjf23/brinla/chicago.html

https://haakonbakka.bitbucket.io/btopic112.html

library(devtools)
## Loading required package: usethis
install_github("julianfaraway/brinla")
## Skipping install of 'brinla' from a github remote, the SHA1 (95181536) has not changed since last install.
##   Use `force = TRUE` to force installation
library(brinla)

library("inlabru")

The default mean and precision for fixed effects are:

inla.set.control.fixed.default()[c('mean','prec')]
## $mean
## [1] 0
## 
## $prec
## [1] 0.001

We see that the default prior on beta is normal with mean zero and precision 0.001. The precision is the inverse of the variance. We convert this to SD:

sqrt(1/0.001)
## [1] 31.62278

We wish to predict the response at a new set of inputs. We add a case for the new inputs (weight= c(45, 40, 65, 31, 53)) and set the response to missing (height=NA):

data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ]

dat <-  data.frame(weight= c(45, 40, 65, 31, 53))

xbar <- mean(d2$weight) 
new_w <- bind_cols(weight, NA)
## New names:
## * NA -> ...1
## * NA -> ...2
colnames(new_w) <- c("weight", "height")

# add the weight values of interest to the dataframe

d1.i <- d2  %>% 
  select(c("weight", "height")) %>%  
  rbind(new_w) %>% 
  mutate(w= weight-xbar) %>% 
  select( c("height", "w"))

#indices of the weights with missing values of height 

d1.i.na <- which(is.na(d1.i$height))

#We need to set the control.predictor to compute the posterior means of the linear predictors
m1.i<- inla(height ~ w, data= d1.i,
            control.fixed = list(
        mean= 0, 
        prec= 1, 
        mean.intercept= 178, 
        prec.intercept= 1/(20^2)),# sd = 20 --> precision =1/variance --> 1/(sd^2)
        control.compute = list(config= TRUE),
        control.predictor=list(compute=TRUE)
)


# posterior means and SDs from the INLA fit

m1.i$summary.fixed[,1:2]
##                    mean         sd
## (Intercept) 154.6013921 0.27107172
## w             0.9034319 0.04200942
# the summary statistics of the fitted values for the weight values of interest can be shown using index of the values with the missing height
m1.i$summary.fitted.values[d1.i.na, ]
##                          mean        sd 0.025quant 0.5quant 0.975quant     mode
## fitted.Predictor.353 154.6100 0.2710729   154.0782 154.6100   155.1417 154.6100
## fitted.Predictor.354 150.0928 0.3426798   149.4206 150.0928   150.7651 150.0927
## fitted.Predictor.355 172.6786 0.8831834   170.9456 172.6787   174.4108 172.6789
## fitted.Predictor.356 141.9619 0.6472126   140.6924 141.9619   143.2316 141.9617
## fitted.Predictor.357 161.8374 0.4320724   160.9897 161.8375   162.6849 161.8375

2.

Model the relationship between height(cm) and the natural logarithm of weight (log-kg): log(weight). Use the entire Howell1 data frame, all 544 rows, adults and non-adults. Use any model type from Chapter 4 that you think useful: an ordinary linear regression, a polynomial or a spline. Plot the posterior predictions against the raw data.

2.rethinking

data(Howell1)
d <- Howell1
d$log_weight <- log(d$weight) 
xbar <- mean(d$log_weight) 

m2 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( log_weight - xbar ) , 
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d )

precis(m2)
##             mean        sd       5.5%      94.5%
## a     138.268410 0.2201360 137.916591 138.620230
## b      47.071138 0.3826317  46.459618  47.682657
## sigma   5.134716 0.1556694   4.885926   5.383506
plot( d$weight , d$height , col=col.alpha(rangi2,0.7) ) 
x_seq <- log(1:60)
mu <- sim( m2 , data=list(log_weight=x_seq) )
mu_mean <- apply(mu,2,mean)
mu_ci <- apply(mu,2,PI,0.99) 
lines( exp(x_seq) , mu_mean ) 
shade( mu_ci , exp(x_seq) )

You could certainly do better—the trend is under-predicting in the mid ages. But just taking the log of weight does most of the work. Why? It’ll help to think of a human body as a cylinder. Roughly. The weight of a cylinder is proportional to its volume. And the volume of a cylinder is: V = πr2h where r is the radius and h is the height. As the cylinder, uh human, gets taller, the radius gets bigger. So we can just say the radius is some fraction α of the height: Substituting that in: r = αh V = πα2h3 = kh3 where k = πα2 is just some proportionality constant.

2.INLA

library(brinla)
data(Howell1)
d <- Howell1

log_wplot <- log(1:60)
new_logw <- bind_cols(log_wplot, NA)
## New names:
## * NA -> ...1
## * NA -> ...2
colnames(new_logw) <- c("log_weight", "height")

# add the weight values of interest to the dataframe

d2.i <- d %>% 
  mutate(log_weight= log(weight)) %>% 
  select(c("log_weight", "height")) %>%  
  rbind(new_logw) %>% 
  mutate(xbar= mean(log_weight),
         log_w= log_weight - xbar)


#We need to set the control.predictor to compute the posterior means of the linear predictors
m2.i<- inla(height ~ log_w, data= d2.i,
            control.fixed = list(
        mean= 0, 
        prec= 1, 
        mean.intercept= 178, 
        prec.intercept= 1/(20^2)),# sd = 20 --> precision =1/variance --> 1/(sd^2)
        control.compute = list(config= TRUE),
        control.predictor=list(compute=TRUE)
)

# posterior means and SDs from the INLA fit

m2.i$summary.fixed[,1:2]
##                 mean        sd
## (Intercept) 137.8410 0.7915664
## log_w        16.3662 1.3418096
#indices of the weights with missing values of height 

d2.i.na <- which(is.na(d2.i$height))

# the summary statistics of the fitted values for the weight values of interest can be shown using index of the values with the missing height
m2.i$summary.fitted.values[d2.i.na, ]
##                           mean        sd 0.025quant  0.5quant 0.975quant
## fitted.Predictor.545  81.98959 4.6919164   72.92648  81.99131   91.16749
## fitted.Predictor.546  93.33351 3.7781530   86.03917  93.33182  100.73021
## fitted.Predictor.547  99.96928 3.2479897   93.70162  99.96524  106.33486
## fitted.Predictor.548 104.67743 2.8750126   99.13237 104.67149  110.31897
## fitted.Predictor.549 108.32935 2.5883628  103.33989 108.32176  113.41437
## fitted.Predictor.550 111.31319 2.3565293  106.77329 111.30413  115.94764
## fitted.Predictor.551 113.83599 2.1627307  109.67221 113.82559  118.09454
## fitted.Predictor.552 116.02134 1.9969785  112.17964 116.00971  119.95964
## fitted.Predictor.553 117.94896 1.8528574  114.38757 117.93621  121.60854
## fitted.Predictor.554 119.67328 1.7260124  116.35833 119.65950  123.08662
## fitted.Predictor.555 121.23311 1.6133539  118.13713 121.21841  124.42929
## fitted.Predictor.556 122.65713 1.5126262  119.75634 122.64160  125.65898
## fitted.Predictor.557 123.96710 1.4221401  121.24154 123.95087  126.79369
## fitted.Predictor.558 125.17994 1.3405932  122.61241 125.16316  127.84862
## fitted.Predictor.559 126.30908 1.2669768  123.88420 126.29188  128.83583
## fitted.Predictor.560 127.36531 1.2004953  125.06907 127.34786  129.76306
## fitted.Predictor.561 128.35750 1.1405103  126.17723 128.33996  130.63807
## fitted.Predictor.562 129.29295 1.0865054  127.21706 129.27553  131.46833
## fitted.Predictor.563 130.17782 1.0380564  128.19454 130.16071  132.25855
## fitted.Predictor.564 131.01729 0.9948077  129.11590 131.00070  133.01253
## fitted.Predictor.565 131.81580 0.9564569  129.98622 131.79993  133.73450
## fitted.Predictor.566 132.57715 0.9227375  130.81048 132.56222  134.42778
## fitted.Predictor.567 133.30465 0.8934081  131.59228 133.29084  135.09521
## fitted.Predictor.568 134.00119 0.8682436  132.33355 133.98868  135.73933
## fitted.Predictor.569 134.66929 0.8470266  133.03897 134.65824  136.36205
## fitted.Predictor.570 135.31118 0.8295415  133.71048 135.30171  136.96498
## fitted.Predictor.571 135.92885 0.8155733  134.35080 135.92105  137.55145
## fitted.Predictor.572 136.52405 0.8049018  134.96144 136.51799  138.12053
## fitted.Predictor.573 137.09835 0.7973052  135.54599 137.09407  138.67545
## fitted.Predictor.574 137.65319 0.7925575  136.10473 137.65068  139.21538
## fitted.Predictor.575 138.18983 0.7904329  136.64044 138.18908  139.74349
## fitted.Predictor.576 138.70943 0.7907078  137.15528 138.71039  140.25852
## fitted.Predictor.577 139.21305 0.7931610  137.64879 139.21564  140.76235
## fitted.Predictor.578 139.70163 0.7975798  138.12503 139.70577  141.25527
## fitted.Predictor.579 140.17604 0.8037609  138.58260 140.18166  141.73729
## fitted.Predictor.580 140.63710 0.8115108  139.02514 140.64407  142.20994
## fitted.Predictor.581 141.08551 0.8206505  139.45208 141.09375  142.67231
## fitted.Predictor.582 141.52197 0.8310132  139.86489 141.53136  143.12570
## fitted.Predictor.583 141.94710 0.8424462  140.26491 141.95754  143.57045
## fitted.Predictor.584 142.36145 0.8548108  140.65287 142.37285  144.00571
## fitted.Predictor.585 142.76558 0.8679822  141.02928 142.77782  144.43303
## fitted.Predictor.586 143.15997 0.8818469  141.39468 143.17297  144.85264
## fitted.Predictor.587 143.54507 0.8963040  141.74967 143.55875  145.26410
## fitted.Predictor.588 143.92132 0.9112637  142.09510 143.93560  145.66763
## fitted.Predictor.589 144.28912 0.9266464  142.43150 144.30393  146.06342
## fitted.Predictor.590 144.64883 0.9423820  142.75946 144.66409  146.45219
## fitted.Predictor.591 145.00081 0.9584076  143.07934 145.01646  146.83406
## fitted.Predictor.592 145.34537 0.9746681  143.39155 145.36136  147.20939
## fitted.Predictor.593 145.68283 0.9911152  143.69641 145.69910  147.57807
## fitted.Predictor.594 146.01347 1.0077063  143.99390 146.02998  147.94029
## fitted.Predictor.595 146.33757 1.0244043  144.28483 146.35428  148.29626
## fitted.Predictor.596 146.65537 1.0411766  144.56953 146.67224  148.64615
## fitted.Predictor.597 146.96711 1.0579945  144.84816 146.98411  148.99008
## fitted.Predictor.598 147.27303 1.0748330  145.12100 147.29013  149.32822
## fitted.Predictor.599 147.57334 1.0916706  145.38825 147.59051  149.66079
## fitted.Predictor.600 147.86823 1.1084882  145.65020 147.88545  149.98798
## fitted.Predictor.601 148.15791 1.1252692  145.90711 148.17516  150.30992
## fitted.Predictor.602 148.44255 1.1419991  146.15918 148.45981  150.62677
## fitted.Predictor.603 148.72232 1.1586657  146.40663 148.73957  150.93866
## fitted.Predictor.604 148.99739 1.1752580  146.64965 149.01461  151.24574
##                          mode
## fitted.Predictor.545  82.0982
## fitted.Predictor.546  93.4060
## fitted.Predictor.547 100.0190
## fitted.Predictor.548 104.7101
## fitted.Predictor.549 108.3482
## fitted.Predictor.550 111.3203
## fitted.Predictor.551 113.8329
## fitted.Predictor.552 116.0092
## fitted.Predictor.553 117.9289
## fitted.Predictor.554 119.6462
## fitted.Predictor.555 121.1999
## fitted.Predictor.556 122.6186
## fitted.Predictor.557 123.9240
## fitted.Predictor.558 125.1332
## fitted.Predictor.559 126.2595
## fitted.Predictor.560 127.3138
## fitted.Predictor.561 128.3049
## fitted.Predictor.562 129.2401
## fitted.Predictor.563 130.1255
## fitted.Predictor.564 130.9664
## fitted.Predictor.565 131.7672
## fitted.Predictor.566 132.5314
## fitted.Predictor.567 133.2625
## fitted.Predictor.568 133.9631
## fitted.Predictor.569 134.6358
## fitted.Predictor.570 135.2826
## fitted.Predictor.571 135.9054
## fitted.Predictor.572 136.5058
## fitted.Predictor.573 137.0855
## fitted.Predictor.574 137.6457
## fitted.Predictor.575 138.1876
## fitted.Predictor.576 138.7123
## fitted.Predictor.577 139.2208
## fitted.Predictor.578 139.7141
## fitted.Predictor.579 140.1929
## fitted.Predictor.580 140.6581
## fitted.Predictor.581 141.1103
## fitted.Predictor.582 141.5503
## fitted.Predictor.583 141.9787
## fitted.Predictor.584 142.3960
## fitted.Predictor.585 142.8028
## fitted.Predictor.586 143.1996
## fitted.Predictor.587 143.5868
## fitted.Predictor.588 143.9650
## fitted.Predictor.589 144.3344
## fitted.Predictor.590 144.6956
## fitted.Predictor.591 145.0487
## fitted.Predictor.592 145.3943
## fitted.Predictor.593 145.7327
## fitted.Predictor.594 146.0640
## fitted.Predictor.595 146.3886
## fitted.Predictor.596 146.7069
## fitted.Predictor.597 147.0189
## fitted.Predictor.598 147.3250
## fitted.Predictor.599 147.6254
## fitted.Predictor.600 147.9203
## fitted.Predictor.601 148.2099
## fitted.Predictor.602 148.4943
## fitted.Predictor.603 148.7739
## fitted.Predictor.604 149.0486
m2.i.postmean <- bind_cols( new_logw[,1], m2.i$summary.linear.predictor[d2.i.na,]) %>% 
  select(c("log_weight", "mean", "sd",  "0.5quant", "0.975quant"))
names(m2.i.postmean) <- c("log_weight", "mean", "sd",  "LCI", "UCI")

m2.i.plot <- ggplot()+
  geom_point(data= d, aes(weight, height))+
  geom_line(data= m2.i.postmean, aes(exp(log_weight), mean))+
  geom_ribbon(data= m2.i.postmean, aes(exp(log_weight), ymin= LCI, ymax= UCI))
m2.i.plot 

3.

Plot the prior predictive distribution for the polynomial regression model in Chapter 4. You can modify the the code that plots the linear regression prior predictive distribution. 20 or 30 parabolas from the prior should suffice to show where the prior probability resides. Can you modify the prior distributions of α, β1, and β2 so that the prior predictions stay within the biologically reasonable outcome space? That is to say: Do not try to fit the data by hand. But do try to keep the curves consistent with what you know about height and weight, before seeing these exact data.